A Chronological Edge-Driven Approach to Temporal Subgraph Isomorphism

Abstract

Many real world networks are considered temporal networks, in which the chronological ordering of the edges has importance to the meaning of the data. Performing temporal subgraph matching on such graphs requires the edges in the subgraphs to match the order of the temporal graph motif we are searching for. Previous methods for solving this rely on the use of static subgraph matching to find potential matches first, before filtering them based on edge order to find the true temporal matches. We present a new algorithm for temporal subgraph isomorphism that performs the subgraph matching directly on the chronologically sorted edges. By restricting our search to only the subgraphs with chronologically correct edges, we can improve the performance of the algorithm significantly. We present experimental timing results to show significant performance improvements on publicly available datasets for a number of different temporal query graph motifs with four or more nodes. We also demonstrate a practical example of how temporal subgraph isomorphism can produce more meaningful results than traditional static subgraph searches.

Sampling Methods for Counting Temporal Motifs

“…Table 2 reports the performance of all algorithms on the 2-node, 3-edge temporal motif in Figure 3A (we chose this motif to allow us to compare against one of the fast algorithms of Paranjape et al [45]). The time span δ was set to 86400 seconds = 1 day in all datasets except EquinixChicago, where δ was 86400 microseconds (these are the same parameters used in exploratory data analysis in prior work [36]).…”

“…The constraints on the algorithm are that it must provide the exact counts along with the so-called duration of the motif instance (the difference in the earliest and latest timestamp in the edges in the motif instance; for example, the duration in the top left motif instance in Figure 1C is 32 -16 = 16). This constraint holds for some existing algorithms [36] but not for others [45]. An additional contribution of our work is a new exact counting algorithm for a class of star motifs that is compatible with our sampling framework.…”

“…Our sampling framework is flexible since it can use any algorithm that exactly counts temporal motifs as a subroutine, provided that this algorithm can be transformed to output the count-duration pairs {(count i , ∆ i )}, where count i is the number of instances of the motif with duration ∆ i . A recently proposed "backtracking" algorithm satisfies this constraint [36]. The fast algorithms for 2node, 3-edge star motifs introduced by Paranjape et al do not satisfy these requirements, since the algorithm uses an inclusion-exclusion rule that cannot output the durations.…”

“…Backtracking algorithm (BT, [36]). The backtracking algorithm examines the edges of the input graph in chronlogical order and matches one edge of the motif at a time.…”

New Algorithms for Counting Temporal Graph Pattern

“…On the other hand, all the edges of δ-temporal motif should be in a fixed time window, while the temporal motif in [27] only requires that the time difference between adjacent edges does not exceed the threshold. Afterwards, Mackey et al [31] used the δ-temporal motif and designed a chronological edge-driven method to search all matched subgraphs of a given temporal graph. Liu et al [32] proposed a sampling framework for counting the δ-temporal motif.…”